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LESSONS IN FRENCH.-XXII.

heure vous éveillez-vous le matin ? 12. Je m'éveille ordinaire.

ment à six heures moins un quart. 13. Vous levez-vous aussitôt SECTION XXXVIII.-USES OF REFLECTIVE VERBS (continued). que vous vous éveillez ? 14. Je me lève aussitôt que je m'éveille. 1. THE reflective verb se passer is used idiomatically in the 15. De quels livres vous servez-vous ? 16. Je me sers des Jense of to do without. It is followed by the preposition de, miens et des vôtres. 17. Ne vous servez-vous pas de cens de when it comes before a noun or a verb.

votre frère ? 18. Je m'en sers aussi. 19. Les plumes dont (Sect. Vous passez-vous de ce livre ? Do you do without that book ? XXX. 8] vous vous servez sont-elles bonnes ? 20. Pourquoi Je ne puis m'en passer, I cannot do without it.

votre ami s'éloigne-t-il du feu ? 21. Il s'en éloigne parcequ'il a 2. Se servir [2, ir. ; see § 62], to use, also requires the prepo- 23. 11 s'en approche pour se chauffer. 24. Vous ennuyez-vous

trop chaud. 22. Pourquoi votre domestique s'en approche-t-il? sition de before its object.

ici ? 25. Je ne m'ennuie pas. Je me sers de votre canif,

I use your penknife.
Je ne m'en sers pas,
I do not use it.

EXERCISE 72. 3. The second example of the two rules above shows that, 1. Will you lend me your penknife ? 2. I cannot do withost when the object of those verbs is a thing, it is represented in it, I want it to mend my pen. 3. Do you want to use my book? the sentence by the pronoun en.

4. I want to use it, will you lend it to me? 5. What knife

does your brother use ? 6. He uses my father's knife, and my Je m'en sers ; je m'en passe, I use it; I do without it.

brother's fork. 7. Will you not draw near the fire ? 8. We 4. The pronoun* used as indirect object of a reflective verb, are much obliged to you, we are warm. 9. Is that young lady if representing a person, follows the verb ($ 100 (4)].

warm enough ? [Sect. XXXIII. 3.] 10. She is very cold. 11. Je puis me passer de lui, I can do without him.

Tell her (dites-lui) to come near the fire. 12. Why do you go Je m'adresse à vous et à elle, I apply to you and to her,

from the fire ? 13. We are too warm. 14. Does your brother 5. S'endormir (2, ir. ; see § 62), to fall asleep, and s'éveiller, leave the window? 15. He leaves the window because he is

cold. 16. To whom does that gentleman apply? 17. He to awake, are also reflective.

applies to me and to my brother. 18. Why does he not apply Je m'endors aussitôt que je me I fall asleep as soon as I go to bed.

to me? 19. Because he is ashamed to speak to you. 20. Do couche, Je m'éveille à six heures du matin, I awake at six o'clock in the morning. you awake early every morning ? 21. I awake early when I go

to bed early. 22. Why do you go to sleep? 23. I go to sleep 6. S'approcher, to come near, to approach ; s'éloigner, to draw because I am tired. 24. Are you afraid to go near your father? back, to leave, take the preposition de before a noun. Their 25. I am not afraid to approach him. 26. Can you do without object, when a pronoun, is subject to Rules 3 and 4 above.

us ? 27. We cannot do without you, but we can do without Votre fils s'approche-t-il du feu ? Does your son draw near the fire ?

your brother. 28. Do you want my brother's horse ? 29. No, Il ne s'en approche pas, He does not come near it.

Sir, we can do without it. 30. Do you intend to do without Il s'éloigne de moi et de vous, He goes from me and from you. money ? 31. You know very well that we cannot do without it. RÉSUMÉ OF EXAMPLES.

32. Is your brother weary of being here? 33. He is not weary

of being here. 34. Come near the fire, my child. Vous servez-vous de ce couteau ?

you use that knife ? Je ne m'en sers pas, il ne coupe pas. I do not use it, it does not cut. De quels couteaux vous servez- What knives do you use?

LESSONS IN ARITHMETIC.-XX. vous ? Nous nous servons de couteaux We use steel lenives.

RATIO AND PROPORTION. d'acier. Pouvez-vous vous passer d'argent? Can you do without money ?

1. In comparing two numbers or magnitudes with each other, Nous ne pouvons vous en passer, We cannot do without it.

we may inquire either by how much one is greater than the Vous passez-vous de votre maitre? Do you do without your teacher ? other, or how many times one contains the other. Nous nous passons de lui. We do without him.

This latter relation-namely, that which is expressed by the Vous adressez-vous à ces messieurs? Do you apply to those gentlemen ? quotient of the one number or magnitude divided by the otherNous nous adressons à eux et à We apply to them and to you, is called their Ratio.

Thus the ratio of 6 to 2 is 6 • 2, or 3. The ratio of 7 to 5 Vous vous endormez facilement. You go to sleep easily. Je m'éveille de très bonne heure. I awake very early.

is 7 = 5, or, as it would be written, the fraction . The two Pourquoi vous approchez-vous du Why do you come near the fire ?

numbers thus compared are called the terms of the ratio. The feu ?

first term called the antecedent, the last the consequent. It Je m'en approche parceque j'ai I come near it because I am cold. will be seen that any ratio may be expressed as a fraction, the froid.

antecedent being the numerator, and the consequent the denoNous nous éloignons du feu. We go from the fire.

minator. A ratio is, in fact, the same thing as a fraction. Nous nous en éloignons. We go from it.

When we talk of a ratio, we regard the fraction from rather a Nous nous approchons de notre We go near our father,

different point of view, namely, as a means of comparing the père.

magnitude of the two numbers which represent the numerator Nous nous approchons de lui. We go near him.

and the denominator, rather than as an expression indicating VOCABULARY.

that a unit is divided into a number of equal parts, and that so Aussi, also. Encre, f., ink.

Ordinairement, gene- many of them are taken. Aussitôt-que, as soon Fenêtre, f., window. rally.

2. The ratio of two numbers is often expressed by writing Feu, m., fire. Plume, f., pen.

two dots, as for a colon, between them. Thus the ratio of 6 to Canif, m., penknife. Fourchette, f., fork. Pourquoi, why. 3 is written 6:3; that of 3 to 5, 3 : 5, etc. Demoiselle, young lady. Heure, f., hour, o'clock. Prêt-er, 1, to lend.

The expressions and 3 : 5, it must be borne in mind, mean Domestique, m., ser. Moins, less, before. Quart, m., quarter.

exactly the same thing. vant. Obligé, -e, obliged. Taill-er, 1, to mend.

A direct ratio is that which arises from dividing the anteceEXERCISE 71.

dent by the consequent. 1. Pouvez-vous vous passer d'encre ? 2. Nous pouvons nous

An inverse or reciprocal ratio is the ratio of the reciprocals* of en passer, nous n'avons rien à écrire. 3. Vous servez-vous de the two numbers. Thus, the inverse ratio of 3 : 5 is the ratio votre plume ? 4. Je ne m'en sers pas ; en avez-vous besoin ? of } : }, or otherwise expressed which is the same as f, or 5. Ne vonlez-vous pas vous approcher du feu ? suis bien obligé, je n'ai pas froid. 7. Pourquoi ces demoiselles otherwise expressed, 5 : 3. s'éloignent-elles de la fenêtre ? 8. Elles s'en éloignent parce

Hence we see that the inverse ratio of two numbers is er. qu'il y fait trop froid. 9. Ces enfants ne s'adressent-ils pas à pressed by inverting the order of the terms when the ratio is vous ? 10. Ils s'adressent à moi et à mon frère. 11. À quelle

* The reciprocal of a number or fraction is the number or fraction The rule does not apply to the reflective pronoun, which is some obtained by inverting it. Thus, the reciprocals of 5, I, d, etc., are times an indirect object.

respectively }, }, 6.

vous.

as.

x 266.

48

expressed by points, or by inverting the fraction which expresses Any set of numbers are said to be respectively proportional to the direct ratio.

any other set containing the same number when the one set can A ratio is said to be compounded of two other ratios when it be obtained from the other by multiplying or dividing all the is equal to the product of the two ratios. Thus, it is a ratio numbers of that set by the same number. Thus, 3, 4, 5 are compounded of the ratios and .

proportional respectively to 9, 12, 15, or to }, , . 3. Proportion.

7. To divide a given number into parts which shall be propor. Different pairs of numbers may have the same ratio. Thus, tional to any given numbers. the ratios , la , are all equal.

Add the given numbers together, and then, dividing the given When two pairs of numbers have the same ratio, the four number into a number of parts equal to this sum, take as many numbers involved are said to form a proportion; and they them- of these parts as are equal to the given numbers respectively. selves, in reference to this relation subsisting among them, are EXAMPLE.—Divide 420 in proportion to the numbers 7, 5, called proportionals. Thus, 3, 4, 12, 16, are proportionals, and 3. because the ratio , or 3 : 4 = the ratio tá, or 12 : 16.

7 + 5 + 3 = 15; A proportion is expressed either by writing the sign of equality and therefore the respective parts are-(=) between the two equal ratios, or by placing four dots in the

it x 420 = 196. form of a square, thus, :: between them.

* x 420 = 140. Thus, the proportionality of 3, 4, 12, 16, might be expressed

it x 420 = 84. in any one of the three following ways :

These parts are evidently in the proportion of 7, 5, and 3, 1 = 18; 3:4 = 12 : 16; 3:4 :: 12 : 16.

and their sum, 196 + 140 + 84 420. The last expression would be read, 3 is to 4 as 12 is to 16. 8. The same method will apply if the given number or

The first and fourth terms of a proportion are called the quantity is to be divided proportionally to given fractions. extremes; the middle two, the means.

EXAMPLE.--Divide 266 into parts which shall be respectively 4. If four numbers be proportional, the product of the extremes proportional to j, , and 1. is equal to the product of the means.

Following exactly the same method as before, the answer, Take any proportion, 3 : 4:: 9:12, for instance. Expressing without reduction, would bem this in the fractional form, we have ins, and reducing these fractions to a common denominator 12 x 4, we get

* 266, and
++

* + + 書
12 x 3 4 x 9
or 12 x 3 = 4 x 9.

Or we may proceed thus :-
48

Reducing the fractions to their least common denominator, Now, 12 and 3 are the extremes, and 4 and 9 are the means,

which is 60, we get of the given proportion.

19, 89, and . Conversely, if the product of two numbers is equal to the pro- Now these fractions are proportional respectively to 40, 45, 48. duct of any other two numbers, the four numbers will form a Hence we have to divide 266 in the proportion of 40, 45, and proportion. Thus, since

48, to which the required answer is, since 40 + 45 + 48 = 133, 8 * 3 = 6 * 4 8, 4, 6, 3 form a proportion;

119 * 266, is's * 266, and it's's * 266, or, 8:4:: 6:3 Or we may write it thus, 8:6 :: 4:3;

or 80, 90, and 96.

EXERCISE 41. or,

3:6:: 4:8;
or, 4:8:: 3:6;

Find in their simplest form :-
or,
4:3 ::8:6.

1. The ratio of 14 to 7, 36 to 9, 8 to 32, 54 to 6. Thus we see that either product may be separated to form 2. The ratio of 324 to 81, 802 to 99. the extremes, and that, the order of either the means or the 3. The inverse ratio of 4 to 12, and of 42 to 6. extremes being interchanged, the numbers still form & 4 Find the fourth term of the proportions, 3:5::6:-; 4:8::9: ~; portion. 5. If three numbers be given, a fourth can always be found

5. Insert the third term in the following proportions-3:5:: - : 6;

4:8::-:9; } :::-:. which will form a proportion with them.

6. Insert the second term in the following proportions-3: -::5:6; This is the same thing as saying that if three terms of a pro- 4:-::8:9; 3:-:::%. portion be given, the fourth can be found.

7. Insert the first term in the following proportions- -:3:: 5:6;

-:4::8:9; - :}:::%. Take any three numbers—3, 4, 5, for instance. Then we have

8. Find a fourth proportional to 2:13, -579, and 3•14159, correct to 3:4 :: 5: fourth term,

5 places of decimals. Therefore

9. Divide 100 in the ratio of 3 to 7.

10. Two numbers are in the ratio of 15 to 34, and the smaller is 75; 3 x fourth term = 5 x 4 (since the products of the means and find the other. extremes are equal).

11. What two numbers are to each other as 5 to 6, the greater of Therefore, dividing both of these equalities by 3

them being 240 ? 5 x 4

As tests by which the correctness of the processes of Fourth term = 5x4 the required number,

addition, subtraction, multiplication, and division may be

ascertained, were given in Lessons in Arithmetic, II. to V., it Here we have found the fourth term, but we could in the has not been thought requisite to give answers to the Exercises same way find a number which would form a proportion with already given in abstract Arithmetic. The answers will, hope the three given numbers when standing in any of the terms. ever, be supplied to future examples in concrete Arithmetic. For instance, for the second term we should have

3 : second term :: 4:5, and therefore

MECHANICS.-IX. 4 X second term = 5 X 3. Hence, dividing both of these equalities by 4

ANOTHER weighing instrument is the steelyard, which (Fig. 54) second term

is a lever of the first order, to the short arm of which is attached

at b a hook from which the substance, w, to be weighed is and similarly for the other two terms.

suspended, while on the long arm slides the movable counterThe most important application of proportion is the solution poise P. The object aimed at in this instrument being that a of examples of this kind, where three terms of a proportion are small weight, P, should balance a large one, w, on the hook, it given to find a fourth. This is what is usually called Rule of is clear that there must be a corresponding disproportion in the Theree, which will be dealt with in a future lesson.

arms—the fulcrum, a, must be near one of the ends of the beam. 6. It is evident that if the two terms of a ratio be multiplied Further, since it is necessary that the steelyard should take an or divided by the same quantity, the ratio is unaltered. horizontal position, both when loaded and unloaded at its hook,

pro

THE STEELYARD.

5 x 3

seven.

it is essential that its own centre of gravity should lie some must be shifted to the point in which F R is to r p in the prowhere on the short arm; for then the counterpoise can balance portion of 16 to 1, there being 16 ounces in the pound. This it when placed in some position on the other arm, such as that comes to dividing the distance e p (which is known) into marked o, in the figure. For this reason steelyards are made seventeen equal parts, as proved in Lesson IV., and taking the heavy at one end.

first point of division next to p for the fulcrum. If there be 2 To Graduate a Steelyard.—The centre of gravity of the beam ounces in the pan, r r must be to FP as 16 to 2; that is, you being on the hook side of the fulcrum, let it be brought into an divide rp into 18 parts, and take the fulcrum 2 from P. If

horizontal position, there be 7 ounces, you divide into 23 parts, and take 7 next to no weight being on P; and so on for all the ounces from 1 to 16 you may determine the hook. Then, as the several positions of the fulcrum, marking them as you proproved in Lessons ceed. If the beam be of any other weight, you follow a similar VII. and VIII., the course, dividing R P into as many equal parts as there are ounces moment of p is equal in the sum of the weights of the beam and substance, and count

to the moment of the ing off as many divisions from p as there are ounces in the 13 12 11 10 9 8 6 5 4 3 2 1

beam, that is, the latter.
weight of the beam From all this it is evident, first, that the subdivisions are not

multiplied into the equal to each other, as in the steelyard ; secondly, that the Р

W

distance of its centre operation of graduation is more troublesome than in that instruof gravity from a ver- ment. The Danish balance, however, has the advantage of not

tical line through the being encumbered with a movable counterpoise; it carries its Fig. 54.

fulcrum, is equal to own imperial standard weight within itself.
P multiplied into the

THE BENT LEVER BALANCE. distance of o from that line. At the point o so found draw a line across the beam ; that line represents the zero division of The principle of this instrument, a species of which is largely the long arm, or the division at which p produces equilibrium, sold for weighing letters, may be understood by the aid of the the weight on the hook being nothing, cipher, or zero.

accompanying Fig. 56. On an upright stand is placed a quadrant Now, supposing that any number of pounds, w, of any sub- arc, M o, of which c is the centre. Round c as a fulcrum revolves stance are hung on the hook, while p is shifted to the left until, a lever, usually bent, but in the figure represented as formed of as in the figure, the arm is again horizontal, we have a multi- two arms at right angles to each other. The arm C B is geneplied by the distance of its ring from the fulcrum a equal to w rally of small weight, being lightly constructed, while the other, multiplied by ab (this line ab being supposed horizontal), together co, called the "index arm, is heavily weighted at its lower with the moment of the beam. But p multiplied by the distance end, the centre of gravity of the whole lever thus being nearly of the zero division from a, is equal to the moment of the beam,

at some point, G, on that arm. as already proved; therefore it follows that p multiplied by its

On some substance, w, to be distance from the zero division is equal to w multiplied by a b.

weighed, being suspended Now, in order to graduate, let us suppose P one pound and w

from B, the index moves from Then we have in numbers seven times a b equal once

its zero point, o, up the quadthe distance of the counterpoise from o, which tells us the exact

rant until the weight of the position of p for 7 pounds on the hook, namely, that you find it by

lever acting at a balances w measuring from o to the left seven pieces each equal to a b.

at B, that is, until the moLet w be 13 pounds or 3 pounds, then in like manner you

ments of these forces are measure 13 or 3 pieces equal to ab. It thus appears that the

equal, which will be when w subdivisions for the successive pounds are equal to each other ;

multiplied by B is equal to and we may therefore lay down the following rule for graduating

the weight of the lever mul. a steelyard :

Fig. 56.

tiplied by G I. The divisions Find first the zero subdivision by bringing the unloaded in

of the quadrant corresponding strument into an horizontal position by the counterpoise. Put to the several weights 1, 2, 3, 4, etc., suspended from B are, then on the hook, or in the pan, such a number of eren pounds however, best determined by experiment for each weight. as will push the counterpoise to the greatest distance it can go on its arm for even pounds, and divide the distance between this

THE LEVER WHEN THE FORCES ARE NOT PARALLEL. last position and the zero point into as many equal parts as In all the cases of levers and weighing instruments we have there are then pounds on the hook. The points of division so so far considered, the forces were supposed parallel-in weighing obtained are the positions of the counterpoise for the several instruments necessarily so. The treatment of the subject is, pounds up to that number.

however, not complete until the condition of equilibrium is deFor half and quarter pounds these divisions must be sub-termined for levers the forces acting on which are not parallel. divided ; and for greater weights than one pound will balance This is the most general case that can occur, and indeed it on the long arm, the counterpoise must be doubled or trebled, includes all the others. To clearly understand it, let a lever be ctc. If the steelyard be intended for weighing small objects, defined a mass of matter of any shape which has one fixed point such as letters, the counterpoise may be ounces, or tenths in it. It may be a bar straight, or simply bent, or bent and of an ounce, or even smaller weights, as occasion requires. twisted, or it may be a solid block. So long as there is one It thus appears that the construction of a steelyard is very point fixed, we may treat it as a lever, that point being the simple, and that any handy person of a mechanical turn fulcrum.

may make one of steel or iron, Moreover, the two forces which act on it are supposed to be

even of a piece of hard such that their directions when produced meet, and that their wood, without much trouble. plane passes through the fulcrum. In cases where the two

forces do not meet, or their plane does not pass through the DANISH BALANCE.

fulcrum, there cannot be equilibrium. For example, the outThis is a species of steel- stretched right arm of a man is a lever, of which the fulcrum is

yard, in which (Fig. 55) the in the right shoulder. Suppose, as he stretches it before him in

R fulcrum is movable, and the a horizontal position, one force is applied to the hand obliquely Fig. 55.

counterpoise is the weight of from him towards the left to the ground, while another acts

the beam acting at its centre horizontally at his elbow towards the right and at right angles cf gravity, P, the substance to be weighed being suspended to the arm; these forces cannot meet, and therefore wonld not from a hook or placed in a pan, at the extremity, R, on under any circumstances keep the arm in equilibrium; further, the other side of the fulcrum. The question is, how may you even were they to meet, they would not so keep it unless their graduate such an instrument ? To do this, let us suppose plane passed through the fulcrum in the shoulder socket. Sapthe beam to weigh 1 pound, and that 1 ounce of some substance posing the forces, therefore, to be as described, namely, that is placed in the scale; then it is evident that the fulcrum, F, their directions meet and their plane passes through the fulornm,

or

THE

Y

Х

A

E

what is the condition of equilibrium ? In order that you may the perpendicular from o on A P. So likewise is the moment clearly understand this, the knowledge of the following geo- of a Q in reference to o equal to a Q multiplied into o y, the metrical principles is necessary.

corresponding perpendicular. What I have then to prove is FURTHER PROPERTIES OF A PARALLELOGRAM AND TRIANGLE. that these products are equal. But they are equal; for, from 1. The area of a triangle is half that of any parallelogram which

Q

the second geometrical has its base for one side, and a line drawn through its vertex parallel

principle above, the to that base for the side opposite.—This

areas of the triangles C appears from Fig. 57, where A v B is the

A OP, A OQ, are half triangle, and A B C D any parallelogram

these products; and, on A B formed by drawing from A and B

by the third, since any two parallel lines A D, BC to meet the

these triangles stand parallel D c to A B through v. For, draw

Fig. 60.

on the common base VE parallel to A D, and therefore parallel

A O, and the line PQ to BC, to meet A B in E. Then the triangle joining their verticos, being a diagonal, is bisected by A R, that is, A V B is made up of the two triangles a VE by that base, their areas are equal. The moments of A P and A , and B V E.

But since A E V D is a parallelo- therefore, in reference to o are equal, as I undertook to prove. gram, the triangle A V E (Lesson III.) is

Now, to apply this to the lever, using the same figure, let us Fig. 57.

equal to AD V, and is therefore half the suppose the two forces to be A P, AQ, meeting, as I have stated

parallelogram A EV D. So likewise is B VE to be necessary, at some point a. Then it is evident, since there half BEVC; and therefore the triangle A V B half A B C D.

is but one point fixed in the body, that there cannot be equi. 2. The area of a triangle is, in numbers, half the product of its librium unless the resultant of AP and a Q passes through that base and the perpendicular from its vertex on that base. This point, and is there resisted by the supports that fix it. The follows from the previous principle. Let the number of inches fulcrum, therefore, you see, must be on the resultant, and thereor feet, say inches, in a B (Fig. 58) be 6, and in the perpen- fore taking o to be the fulcrum, we must have a P multiplied into dicular, y E, be 7, and construct on A B a parallelogram, ex equal to a q multiplied into o y, that is, ABCD, whose sides are parallel to this perpendicular. Such a

the moments of the forces in reference to parallelogram is termed a “rectangle,” on account of its angles the fulcrum must be equal. We arrivo being all right angles. Mark out the inches

thus at the two following modes of stating on A B and v E, and draw the dotted lines in e

the condition of equilibrium in a lever, the figure parallel to A B and v E, cutting

either of which may be selected for use as this rectangle into the smaller ones the sides

the occasion requires :of which are all equal to one inch, and which

1. In a lever, the forces not being parallel, are therefore so many square inches. Now

the power multiplied by the perpendicular there are seven rows of these squares, one row

from the fulcrum on its direction is equal above the other, and there are six squares in

to the resistance multiplied by the perpen

Fig. 61.

dicular on its direction. each row; and therefore there are altogether 7 times 6, or 42, square inches in the rectangle.

2. The power and resistance are to each other inversely as the But the triangle being half the rectangle, is

& perpendiculars dropped from the fulcrum on their respective half of 42 square inches, that is, it is, in num

Fig. 58.
directions.

THE WHEEL AND AXLE.
bers, half the product of the base and perpen-
dicular. Were the numbers 13 and 9, or any other pair whatever, Figs. 61, 62,

and 63, is a kind of lever, or succession of levers,

This useful mechanism, of which several forms are given in the reasoning would be the same.

3. If two triangles stand on opposite sides of a common base, revolving round an axis, from which they project at right angles. and the line joining their vertices is bisected by that base, the

Corresponding to this central axle triangles have equal areas.-In Fig. 59, the triangles A B C, ABD

line is a cylindrical axle of some stand on the common base, A B, at opposite sides, and Dc join

thickness, round which winds the ing their vertices is supposed to be bisected at m; I have to

rope which bears the resistance, or prove that the areas of the triangles are equal. Draw E F and

weight, to be raised. In Fig. 61 is Hg through A and B parallel

the simplest form of the instrument, G to DC, and also through Dand

consisting of an horizontal axle and c draw H E and G F parallel

four levers, which are worked in to AB Then we have a

succession by the power. In the large parallelogram E F G H,

ship’s capstan for raising the anchor which is divided into four

Fig. 62.

(Fig. 62), the resistance acts horizon. smaller ones by A B and D c.

tally, a man pushing also horizontally But since DC is bisected at the end of each lever, the power being multiplied in the pro

at M, making mc equal to portion of the number of levers and men. We have in Fig. 63 Fig. 59.

MD, and therefore A E equal another form, where the levers are the spokes of a wheel, and the

to A F, the parallelograms power A works in succession on them along above, the triangles A B C and A B D aro half of these parallelo- the resistance and power be parallel or not, AFGB and A E H B are equal to each other. But, as proved the tire as they come round.

The principle in all is the same, whether grams, and therefore are also equal to each other, as was required to be proved.

and may be understood from Fig. 64, which We now return to our Mechanics, applying these geometrical represents a transverse section, the outer principles to determine

circle being the wheel and the inner the

axle. The central line of the axle, which THE MOMENTS IN THE LEVER OF FORCES NOT PARALLEL.

you must conceive perpendicular to the Two such forces, A P, A Q (Fig. 60), being supposed to meet at paper at the centre of these circles, is the some point, A, to which they are transferred, and there com- fulcrum, represented by the point o. The pounded into a resultant A R, represented by the diagonal of the line A B thus is seen to be the lever, at the parallelogram, A PR Q, and o being a point taken at random on ends of which the power, P, and resistance, that diagonal, we can prove the following proposition :

w, act; and, as already proved, these forces

Fig. 63. The moments of two intersecting forces in reference to any point must hr inversely as o A to o B, which lines on their resultant are equal to each other.—Now the moment of a are the radii of the wheel and axle respectively. When the power force in reference to a point, as has been already explained, is and resistance act parallel to each other this is evident; but the the product of the force by the perpendicular dropped on it from same holds good were they not so to act, as in the capstan, where that point. In Fig. 60, therefore, the moment of A p in refe- the power is continually changing direction as the sailors go round; rence to o, a point on the resultant, is A multiplied into ox, for, referring again to Fig. 64, if the power were to act not in the

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B

W

line A P, but along any other tangent to the large circle, the per.

LESSONS IN GERMAN.-XXI. pendicular from the fulcrum o on its direction would still be the radius of the wheel; and, by the general principle of the lever

SECTION XL.-PECULIAR IDIOMS—(continued). established in this lesson, the power and Was für ein ($ 66.5), literally, what for a, answers to the English resistance would be still inversely as the

“what kind of," or simply "what;" as :-Was für ein Buch haben radii of wheel and axle.

Sie? what kind of a book have you? Wag für ein Meffer ist das? A treadmill, used for punishment in prisons, what kind of a knife is that? Für, this connection, loses its is another instrument of this kind, the power prepositional character, and may precede any case, as :—Was für being the weight of the prisoners ascending Vücher sind dies? what kind of books are these ? Was für Bücher the steps placed on the outside of the wheel, haben Sie? what kind of books have you ? Mit was für einem and the resistance the weight of the water Buche sind Sie beschäftigt? with what kind of (a) book are you em. pumped, the corn ground, or other work done.

ployed ? P

The windlass is another, turned generally by 1. Was für is likewise used in the way of exclamation, corre. Fig. 64,

a winch handle, and used to raise water from sponding to "what,” as :—Was für Thorheit! what folly! Was

wells, or lift goods into stores. In Fig. 21 für ein Mann! what a man! Welch, abbreviated from welcher, is (page 188) the reader will find an example of the utility of the used in the same manner, as:—Welch ein Mann! what a man! wheel and axle as a mechanical power in the crane, by which two 2. Jeter and jeglicher are often preceded by the indefinite article, men, by turning the winch-handle attached to the axle, are able and are then, accordingly, inflected after the Mixed Declension. to lift a horse out of the steamor alongside of the quay.

(Sect. X.) They are never used in the plural, as:- Der Tod jetet A particular form of the windlass, which was first invented in Menschen, or cines jeden Menschen ist gewiß, the death of every man is China, and which may therefore be called the “Chinese windlass,” certain. Ein Ieder muß sterben, every one must die. is given in Fig. 65, where only the axle is represented, consisting 3. Aller, unlike the English “all," is joined directly to its of two parts, one thicker than the other, but both forming one noun without any article intervening, as:-Aller Wein, all the solid piece. The winch handle,

wine. Alles Wasser, all the water, etc. or wheel, is to the right con

Our word "all,” when connected with the names of countries, nected with the larger axle. The

towns, etc., as also in such phrases as “all day, all the time, weight to be raised is suspended

all my life," etc., is not expressed in German by all, but by from a hook attached to a pul.

ganz, as :-Ganz Europa, all Europe. Ganz Böhmen, all Bohemia. ley, round which the lifting rope

Die ganze Schweiz

, all Switzerland. Den ganzen Tag, all the day, or passes, one part winding round

the whole day. Die ganze Zeit, mein ganzes Leben, eto. the thick axle while the other

Alle or all, in some elliptical phrases, is equivalent to our "all unwinds from the thin. The

gone,” "no more," and the like, as : -Sein Gelt ist alle, his money weight with each turn of the

is all gone. wheel ascends by the difference

4. Mancher answers to "many a,” as :-Mancher Reiche ist un between the length of the rope

glüdlich, many a rich man is unhappy. that winds and unwinds, that

5. Solcher is often preceded by the indefinite article, as also is, by the difference between

by fein, and is then, like jeder and jeglicher, inflected after the the circumferences of the

Mixed Declension, as :-

-- Er ist eines solchen Lebens nicht würdig, he is two axles. Moreover, since

not worthy of such a (a such) life. Ich Habe fein solches Buch, I the weight is equally divided

have no such book. between the two ropes which ascend from the pulley, the

Fig. 65.

6. Aller, mander, solcher (and welcher, see R. 1) often drop the

last syllable, and are then undeclined. Thus, aller, when it preforce acting at the circumference of each axle is half the weight. cedes a pronoun, is often abbreviated to all; mancher

, when it It is evident, moreover, that the power applied to the winch precedes an adjective, often becomes mandh; solcher (as also handle has to balance the difference of the actions of these welcher) is always thus abbreviated when it precedes the indefiforces at the axle, or the moment of the power must be equal to nite article, as also, sometimes, when it recedes an adjective, the difference of the moments of these foroes. But each force - Ich habe all mein Geld verloren, I have lost all my money. being half the weight, its moment is half the weight multiplied Ich habe all diere Bücher gekauft, I have bought all these books. by the radius of the axle at which it acts; and therefore their Manch ehrlicher Mann ist arm, many an honest man is poor. difference is equal to half the weight multiplied by the difference solch ein Tag ist angenehm, such a day is agreeable. Selo of the radii of the axles, or, which comes to the same thing, schönes Papier ift theuer, such beautiful paper is dear, etc. It to the weight into half the difference of these radii. But the should however be noted, that, as in the above examples, when moment of the power being that force into the radius of the the abbreviated form is followed by an adjective, this latter, inwheel, we immediately learn that,

stead of being infected after the New Declension (Sect. IX. 2), In the Chinese windlass the power multiplied follows that of the Old ($ 29). by the radius of the wheel is equal to the resist- 7. Giniger and etlicher are regularly declined. They are ance multiplied by the difference of the radii nearly synonymous, and answer to our words "some, a fer," of the axles.

etc., as :-&r sprach nur einige Worte, he spoke only a few words. THE COMPOUND WHEEL AND AXLE.

Er hat noch etliche Freunde in Deutschland, he has still some friends

in Germany. Er wohnt in einiger Entfernung von der Stadt, he reThis is a combination of wheels and axles, of sides at some, or a little distance from the city. Nach einiger Zeit the kind already explained, made for the same fam er, after some time he came. Ich habe noch etliches Mehl

, I still purpose as the similar combination of levers in have got some flour. Etliches fiel an den Weg, some fell by the way. Lesson VIII., namely, the mechanical advan- side (Mark iv. 4). tage of a multiplication of the effect of the

8. Gowns, besides the signification noticed in Sect. XIV. 2, has power. The wheel and axle being once clearly also an adverbial use, and answers to “ somewhat," as :-Er ist understood to be a lever, there can be no etwas älter, als ich, he is somewhat (or something) older than I.

difficulty in extending the rule which holds Es ist etwas fälter, als vorgestern, it is somewhat colder than the day Fig. 66.

good of the compound lever to this combina before yesterday.
tion. In Fig. 66 is such a combination. By

VOCABULARY. cogged teeth the axle of each wheel works on the circumference of the next succeeding, the power, P, being applied by a rope to Abʻlegen, to lay aside. Beschwer'te, f. hard. Blind, blind. the circumference of the first wheel, which does not require teeth. Anblid, m. aspect. ship.

Darü'ber, about it, It is evident that, as explained of the compound lever, the condi- An'näherung, f. ap- Vesip'en, to possess. thereon. tion of equilibrium must be that

proach.

Berun'derung, f. ad. Davon', of it, tharsIn the compound wheel and axlo, the power is to the resistance Bege'hen, to commit. miration.

of. as the product of the radii of the axles is to the product of the Beschäf'tigen, to em. Bilden, to form, con. Dennocy, still, not radii of the wheels.

ploy. stitute.

withstanding,

as :

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